From Encyclopedia of Mathematics - Reading time: 1 min
and over a ring
Matrices such that can be transformed into by a sequence of elementary row-and-column transformations, that is, transformations of the following three types: a) permutation of the rows (or columns); b) addition to one row (or column) of another row (or column) multiplied by an element of ; or c) multiplication of a row (or column) by an invertible element of . Equivalently, is obtained from by multiplication on left or right by a sequence of matrices each of which is either a) a permutation matrix; b) an elementary matrix; c) an invertible diagonal matrix.
Equivalence in this sense is an equivalence relation.